What constants $a_1,a_2, \cdot \cdot \cdot, a_n \in \Bbb N $ make the following integral equal to $1$?
$$ \int_0^1 \frac{a_1e^{\frac{1}{\log(x)}}}{x\log^k(x)}+ \frac{a_2e^{\frac{1}{\log(x)}}}{x\log^{k-1}(x)}+\cdot\cdot\cdot+\frac{a_ne^{\frac{1}{\log(x)}}}{x\log^2(x)}~dx$$
I found that for $a_1, \cdot \cdot \cdot, a_3$ being equal to $1,2,1$ the integral is $1.$ Also for $a_1, \cdot \cdot \cdot, a_5$ being equal to $1,6,8,4,1$ the integral is $1.$ I'm trying to see if the sequence follows some pattern.